What are quantum computers?

Quantum computers compute with quantum states using the subtleties
of quantum mechanics to, for suitable problems, do more than we
know how to do with classical computers. They are particularly
beneficial for simulating other quantum systems such as problems
in chemistry and biochemistry where the electrons that form
chemical bonds behave quantum mechanically.

That's a big deal: Unlike classical computers that struggle to do
in silico (by computers) what is relatively easy to do in vitro
(in a lab experiment), quantum computers will excell at computing
even what is hard to do in the lab. That includes things like
designing proteins for tasks where mother nature does not provide
a template: It goes far beyond even the big dreams genetic
engineering fuels, it levels that up to become highly targetted
genetic engineering. Imagine fuel plants that are truly climate
neutral by igrowing a tap for alcohol fuel, or retrofitting human
cells to combat effects of aging: Quantum computers can be the
solution to the really big challenges and ambitions.

Would you like to know how quantum computers work? Then read on!

How do Quantum Computers Work?

What is a Qubit and What can one do with it?

A qubit or quantum bit is the smallest unit of information that a
quantum computer can operate on. It consists of two states which
one calls state 0 and state 1 (or |0> and |1> using a common
notation named after Nobel Laureate Paul Dirac).

A qubit is more than a regular bit. A bit can only ever be in one
state, either 0 or 1. A quantum bit can be in any superposition of
these states, meaning that it can (but does not have to be) in
both states simultaneously.

That's not magic but quantum mechanics: Every state represents
some kind of wave (which we don't necessarily have to imagine as
such). But if you know waves, you know that they have amplitudes
(their height) and phases (the timing where they are on the way
between crest and trough). And it all goes in circles, like any of
the following ones:

troughcrestneutraloppositeneutral

-i |0〉i |0〉-|0〉|0〉

-|1〉|1〉-|0〉|0〉

Just like a wave can be pictured as a rolling wheel (first
picture), states can be changed like rotating a circle (next two
pictures). The difference between the two pictures relating to
states are the directions in which we turn what really is not a
2-dimensional circle but the surface of a 4-dimensional sphere to
which the circles belong.

That is all one can do with a single qubit: Rotate it. In fact,
that is all one can do with a number of qubits, except that the
dimensions along which one can rotate become huge quickly: They
grow exponentially in the number of qubits.

What is a Quantum Gate and how does it act?

The rotations that can be done to qubits have a name: They are
called quantum gates in analogy to the circuit elements, or gates,
that form classical computers. Quantum gates are drawn as a box
with a letter or other description inside. The qubits used as
input and output are drawn as horizontal lines from left to
right. Everything together is a quantum circuit, such as this:

|0〉H

This quantum circuit is called a (true) random number
generator. It does something classical, digital computers cannot
really do because, given a definite input, they produce a definite
output. So how does this simple quantum circuit manage to one-up
the strict definition of a classical, digital computer?

Reading from left to right, the quantum circuit says to start with
a qubit initialized to the state |0> and to feed it into a H
gate. The H gate is named after the mathematician Jacques Hadamard
and hence often called Hadamard gate. It says to rotate a qubit
such that an input state |0> ends up half-way between that state
and the |1> state. Finally, the rightmost symbol in the quantum
circuit instructs a quantum computer to measure if the qubit is in
the |0> or in the |1> state.

Measurements are to this day hard even for quantum physicists and
have started big debates about "interpretations" of quantum
dynamics. But the basic math behind them is clear: The outcomes
|0> and |1> occur with a probability proportional to the square of
the abscissa of the point on the circle that corresponds to the
input state. In our case, the abscissas are equal and so are the
probabilities: Our quantum computer outputs |0> or |1>, each with
the probability one half.